Even and Odd Functions

evenoddheadfigThe concepts of even and odd functions are usually introduced in advanced high school algebra, college algebra, trigonometry, or precalculus courses. Trig students learn how to apply the concepts of even and odd functions to simplify trigonometric expressions. Calculus students learn how to apply the concepts of even and odd functions to simplify the calculation of a definite integral. When students understand and can recognize even and odd functions, they are usually amazed how often these functions appear in application problems.

This post will discuss even and odd functions from both an algebraic and geometric point of view. Readers are encouraged to download my free handout Even and Odd Functions which is a handy reference that teachers can give to students.

I will start the discussion by describing even functions. The text box below gives a description of an even function from several points of view. Graph A illustrates what we mean when we say that a graph has symmetry with respect to the y-axis.

evenoddtxtbx1

evenoddfigA

 

The text box below gives a description of an odd function from several points of view. Graph B illustrates what we mean when we say that a graph has symmetry with respect to the origin.

evenoddtxtbx2

evenoddfigB

 

The text box below gives some basic observations about even and odd functions. My free handout Even and Odd Functions gives a more in depth list of the important properties of these functions.

evenoddtxtbx3

 

The text box below shows the infinite Taylor series expansion of the function y = Cos(x). Graph C shows the graph of y = Cos(x) and the graph of the first five terms of the Taylor series expansion of Cos(x). Notice that the Taylor series expansion of Cos(x) is the sum and difference of even functions!

evenoddtxtbx4

evenoddfigC

 

The text box below shows the infinite Taylor series expansion of the function y = Sin(x). Graph D shows the graph of y = Sin(x) and the graph of the first five terms of the Taylor series expansion of Sin(x). Notice that the Taylor series expansion of Sin(x) is the sum and difference of odd functions!

evenoddtxtbx5

evenoddfigD

 

I will end this post by showing you how to create an even or odd function from any function y = f(x) that’s not necessarily even or odd. The text box below shows how and why this can be done. Graph E shows the results of creating an even function and an odd function from the function y = f(x) = 0.25(x – 4)2 – 3Sin(x – 4) – 5. This result explains why the hyperbolic functions cosh(x) and sinh(x) are even and odd functions respectively.

evenoddtxtbx6

evenoddfigE

 

More useful tools from Math Teacher’s Resource

•   The graphs in this post were created with my software, Basic Trig Functions. I think that you will find it very useful for teaching mathematical concepts in your classroom and developing custom instructional content. Check it out at mathteachersresource.com/trigonometry.

•   In addition to the Even and Odd Functions Handout linked in this post, Math Teacher’s Resource offers a wide variety of free math handouts, lessons, and exercises available at mathteachersresource.com/instructional-content. All content is available for immediate download. No sign-up required; no strings attached!

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